R     PA     P  P  A 


777 


MATHEMATICS  ?,!5 
PAPER  LOCATION 
OF  A  RAILROAD 


J,  C.  L.  FISH 


MATHEMATICS 


OF   THE 


PAPER  LOCATION 
OF  A  RAILROAD 


J.   C.   L.    FISH 

Assoc.  M.  Am.  8oc.  C.  E. 

Associate  Professor  of  Civil  Engineering 
Leland  Stanford,  Jr.,  University,  Palo  Alto,  Cal. 


NEW  YOKK 

M.     C.    OLAEK 

13-21  PARK  Row 

1905 


Copyright,  1905,  by  M.  C.  Clark 


The  object  of  this  article  is  to  present  the 
mathematical  steps  involved  in  preparing,  from 
the  paper  location  of  a  railroad,  a  set  of  mathe- 
matically consistent  alinement  notes  by  which  to 
run  the  corresponding  field  location;  and  to  sug- 
gest an  orderly  arrangement  for  the  necessary 
computations. 

Mr.  W.  B.  Storey,  Jr.,  in  discussing  a  paper  by 
Michael  L.  Lynch  on  Railroad  Location,  says:* 

On  the  Southern  Pacific  System  the  location  ia  trans- 
ferred from  the  paper  to  the  ground,  not  by  scaling,  but 
by  calculation.  Each  tangent  is  fixed,  and  the  connect- 
ing curves  are  all  calculated  in  the  office  by  carrying  the 
line  from  one  fixed  tangent  around  through  the  prelim- 
inary to  the  next  tangent.  These  calculated  notes  are 
then  given  to  a  machine  known  as  the  locating  party 
and  put  on, the  ground  by  it. 

The  writer  does  not  remember  having  seen  else- 
where any  reference  to  the  calculation  of  field 
notes;  but  in  various  articles  on  railroad  location 
he  has  noted  direct  or  implied  reference  to  scal- 
ing. 

To  make  clear  what  is  meant  by  scaling  and  by 
calculating,  and  to  show  the  advantage  of  calcu- 
lating over  scaling,  let  us  find,  by  the  two 
methods,  the  station  and  "plus"  via  the  prelimi- 
nary and  via  the  location  for  the  check  point  A, 
Fig.  1,  the  first  crossing  of  the  location  and  pre- 
liminary w;hich  have  a  common  starting  point,  Pi. 

BY  SCALING.— The  angle  at  Pi,  between  pre- 
liminary and  location,  is  scaled  in  the  course  of 
preparing  the  notes  for  field  location.  Simply  for 
the  purpose  of  getting  a  check  on  the  location  at 

•Trans.  Am.  Soc.  C.  E.,  Vol.  XXXI.,  p.  92. 

(Reprinted  from  Engineering  News, 
March  16,  1905.) 


381115 


5000  FT. 


4000  (1500,4000)  R 


1000 


Pape 


(3000, 4200)  fg 

<fr£2f£.'/r 


P7  (4000, 4-500) 


4000  FT. 


FIG.  1.     METHOD  OF  PLOTTING  PAPER  LOCATION  OF  A 
RAILWAY  LINE. 


the  point  A  we  scale  the  distance  PiA  on  the  loca- 
tion and  the  distance  PaA  on  the  preliminary,  thus 
obtaining  the  station  and  plus  for  A  via  each  of 
the  two  lines.  We  now  have  of  the  triangle 
PiPaA  the  given  side  PiPa  (taken  from  the  field 
notes  of  preliminary),  the  given  angle  PiPaA,  the 
two  scaled  sides  PiA  and  PaA,  and  one  scaled 
angle,  APiPa.  All  the  scaled  quantities  are  af- 
fected 'by  the  errors  of  scaling,  which  are  large 
compared  with  field  errors,  and  it  is  evident  that 
our  values  for  the  five  parts  of  the  triangle  are 
mathematically  inconsistent,  since  any  three  parts 
(which  include  one  side)  of  a  triangle  determine 
the  other  parts.  The  result  is  that  when  the  field 
party  has  located  to  A,  and  found  they  do  not 
check  on  the  preliminary  within  several  feet,  they 
have  no  means  of  telling  how  much  of  the  error  is 
due  to  surveying  and  how  much  to  scaling.  •  In 
this  case  the  surveying  checks  the  scaling,  but 
there  practically  is  no  check  on  the  field  work. 

BY  CALCULATION.— iBy  this  method  we  scale 
only  so  many  parts  of  the  triangle  PiPaA  as  will, 
together  with  the  given  part  or  parts,  determine 
the  size  and  shape  of  the  triangle.  Side  PiPa  and 
angle  PiPaA  are  known  from  the  preliminary 
notes.  If,  then,  we  scale  the  angle  PaPiA  we  shall 
have  values  for  three  parts  of  the  triangle,  from 
which  we  can  compute  the  sides  PiA  and  PaA. 
These  five  parts  of  the  triangle — 'two  given,  one 
scaled  and  two  computed — 'are  mathematically 
consistent;  and  when  the  location  party  arrives  at 
A  and  find  they  do  not  exactly  check  on  the  pre- 
liminary, they  know  that  the  error  is  all  charge- 
able to  field  work  (assuming  that  no  errors  have 
been  made  in  computing).  In  this  case  the  field 
work  is  really  checked.  Of  course  the  error  may 
be  in  running  the  preliminary  or  the  location,  or 
more  probably  in  both.  The  check  is  on  the  sur- 
veying on  both  lines  between  Pi  and  A.  If  the 
error  is  not  within  the  limit  of  error  permitted, 
the  location  is  re-run  (using  the  same  notes),  and 
if  this  is  found  to  be  practically  correct,  the  next 
step  is  to  re-run  the  preliminary  from  Pi  to  A.  If 
the  error  is  not  discovered  here  it  must  be  found 
in  the  office  work — computations  or  copying  of 
notes.  It  cannot  be  due  to  scaling. 

5 


MATHEMATICS  OF  PAPER  LOCATION: 
USING  RECTANGULAR  CO-ORDINATES.— The 
chief  points  of  the  preliminary,  surveyed  by  any 
method  or  combination  of  methods,  have  been 
plotted  by  rectangular  co-ordinates,  and  our  map 
shows  points  Pi,  P2,  P3,  P4,  Ps,  Pe  and  PT.  The  co- 
ordinates of  each  point  are  written  by  the  point, 
the  value  of  the  abscissa  first.  The  engineer  has 
drawn  in  pencil  on  the  map  the  location  Pi,  PCi, 
PTi,  PCs,  PT2,  PCs.  PCi  and  PTi  are  respectively 
the  beginning  and  end  of  a  6-degree  curve;  and 
PCz  and  PTa  are  the  beginning  and  end  respec- 
tively of  an  8-degree  curve.  PCs  is  the  beginning 
of  a  curve,  and  is  considered  the  end  of  this  loca- 
tion. It  will  be  noticed  that  each  curve  is  num- 
bered, and  that  each  letter  standing  for  a  curve 
element  is  given  for  a  subscript  the  number  of 
its  tmrve;  e.  g.,  the  central  angle  of  curve  No.  2  is 
AS.  All  dimensions  which  appear  on  the  map  at 
this  time  are  put  in  parentheses  on  our  figure  in 
order  to  distinguish  them  from  the  quantities 
computed  in  the  work  of  getting  out  the  location 
notes,  which  are  written  without  parentheses. 
We  have  nothing  to  do  here  with  the  question  of 
the  proper  place  for  the  location  line  on  the  map: 
that  is  an  engineering  problem,  while  our  work  is 
to  take  the  location  line  as  we  find  it,  and  get  out 
the  notes  for  transferring  it  to  the  ground.  Hav- 
ing no  need  of  the  topography  in  what  follows,  It 
has  been  omitted  from  our  map.  The  distances 
have  been  computed  here  to  the  nearest  foot  only, 
although  it  is  customary  to  consider  tenths  of  a 
foot.  Of  course,  in  any  given  location  the  limit  of 
error  in  field  work  should  control  the  precision  of 
computation.  To  save  space  there  have  been 
omitted  from  Fig.  1  the  station  numbering  of  the 
preliminary  and  of  the  location. 

1.  PRODUCE   THE   TANGENTS   to  intersect 
at  Vi  and  V2. 

2.  SCALE  THE  CO-ORDINATES  of  the  initial 
point,  the  final  point,  and  of  all  the  points  of  tan- 
gent intersection.    In  our  case,  the  initial  point  of 
the  location  is  at  the  origin  of  co-ordinates,  and 
coincides  with  the  initial  point  of  the  preliminary. 
The  co-ordinates  of  Vi  and  V2  and  PCs  are  written 
by  the  points  on  the  map  as  they  are  scaled  off, — 
x  first  and  y  second. 

6 


No  further  scaling  is  done,  except  that  every 
computed  distance  and  angle  should  be  roughly 
checked  by  scaling.  We  are  now  ready  to  take  up 
the  computing. 

3.  COMPUTE  THE  BEARINGS  AND 
LENGTHS  OF  TANGENTS  PRODUCED  TO 
INTERSECTION.— The  plus  direction  of  the 
y-axis  is  taken  as  north.  The  tangent  of  the 
bearing  of  PiVi  is 

tan  *SPV  =  XY,-XP,   „  3,zso~o 


(Throughout,  the  subscript  of  an  x  or  a  y  of  any 
point  is  the  symbol  which  designates  that  point; 
and  the  subscript  of  £  is  composed  of  the  two 
symbols  which  designate  the  terminal  points  of  a 
line.) 


log  3,250    —3.51188  log    3,250=3.51188 

log  1,610    =  3.20683  log  sin  63°  39'  =  9.95236 

log  tan  63°  39'  •=  0.30505  log    3,627  =  3.65952 

From    the   logarithmic   computation    we   find    the 
bearing  and  length  of  PiVi  to  be  N  63*   39'  E, 
3,627  ft. 
The  tangent  of  the  bearing  of  ViVa  is 


,        3,640-3,250        2,030 

and 


log  —  2,100  =  3.32222  n        log  2,100  =  3.32222 
log  2,080  =  3.30750      log  sin  46^  58'  -  9.85609 

45°  58'  =  0.01472  n  2,921  =  3.46553 

That  is,  the  bearing  and  length  of  ViVi  are 
N  45°  58'  W,  2,921  ft.  (The  negative  value  of 
the  tangent  shows  that  the  bearing  is  either  S  E 
or  N  W,  but  the  map  indicates  that  the  bearing 
is  NW.) 


The  tangent  of  the  bearing  of  VaPCs  is 


and 


log    2,380  =  3.37658  logr    2,380  —  3.37658 

log        480  =  2.68124  log  sin  78°  36'  =  9.99135 

log  tan  78°  36'  =  0.69534  log    2,428=3.38523 

giving-    for    VaPCa    the    bearing    and    length      of 
N  78°  36'  E,  2,428  ft. 

We  now  scale  the   map  for  a  rough  check  on 
these  computed  bearings  and  distances. 

4.  COMPUTE       CENTRAL,       ANGLES         OF 
CURVES.—  It  is  evident  from  the  map  that    the 
central  angle  for  curve  No.  1  is 

Ai  =  ^P]VI  +  ^v!V2  =  63°  39'  +  45°  58/  =  109°  37/  ; 
and  the  central  angle  for  curve  No.  2  is 

Aa  •=  ^Vj  +  ^V2  PC3  =  45°  58'  +  78°  36/  =  124°  34/- 
We  roughly  check  these  values  by  scaling  the 
map  with  the  protractor. 

5.  COMPUTE     TANGENT  -  DISTANCES     OP 
CURVES.—  Using   a   "table   of    tangent-distances 
for  a  1°.  curve"  we  find  the  tangent-distance  for 
curve  No.  1  is 

Ti  =  1,355; 
and  for  curve  No.  2, 

T2  =  1,364. 

We  obtain  a  rough  check  on  these  by  scaling. 
If  in   any  case   the   value   of    A   is   beyond   the 
limits  of  the  table,  of  course  the  tangent-distance 
must  be  computed  from  the  formula: 
T  =  R  tan  A/2. 

6.  COMPUTE  CURVE    LENGTHS.—  Length    of 
curve  is  equal  to  one  hundred  times  the  ratio  of 
central   angle  to   degree  of  curve. 

Length  of  curve  No.  1  is 

Lj  »  100  Ai  /  I>i  =  100  (109°  370/6°   =  100  (109.62)  /  6 
=  1,827. 

8 


Length  of  curve  No.  2  is 

Ls  —  100  Aa/Da  —  100  (124°  34')  /  8°  =  1OO   (124.57)  /  8 
—  1,557. 

We  now  step  off  the  curve  lengths  on  the  map 
to  obtain  a  rough  check  on  the  computed  values. 

7.  COMPUTE    LENGTHS    OF    CENTER-LINE 
TANGENTS.— The  length  of  the  first  tangent,  as 
the  map  shows,  is 

PiPCi  =  PiVi  —  Ti  =  3,627  —  1,355  =  2,272. 
The  second  tangent  is 

PTi  PC2  =  ViVs  —  (Ti  +  T2)  =  2,921  —  (1,355  + 
1,364)  =  202; 
And  the  third  tangent  is 

PTa  PCs  =  Va  PCs  —  T2  =  2,428  —  1,364  =  1,064. 
The  rough  check  by  scaling  is  now  employed. 

8.  COMPUTE      STATION     AND     PLUS     FOR 
CURVE  AND  TANGENT  POINTS.— The    station 
and  plus  for  Pi  is  0  +  00. 

The  station  and  plus  for  PCi  Is  PiPCi  = 
(22  +  72). 

The  station  and  plus  for  PTi  is  (22  +  72)  +  La.  = 
(22  +  72)  +  1,827  =  (40  +  99). 

The  station  and  plus  for  PCs  is  (40  +  99)  +  PTi 
PCs  =  (40  +  99)  +  202  =  (43  +  01). 

The  station  and  plus  for  PTa  is  (43  +  01)  +  L§  = 
(43  +  01)  +  1,557  =  (58  +  58). 

The  station  and  plus  for  PCs  is  (58  +  58)  +  PT« 
PCs  =  (58  +  58)  +  1,064  =  (69  +  22). 

Now  we  prick  off  the  stations  on  the  map,  thus 
checking  roughly  these  values. 

9.  COMPUTE      ELEMENTS      FOR      CHECK 
POINTS. — On   our  map   the   location   crosses   the 
preliminary  at  the  point  A,  which  we  use  as  a 
check  point.    There  is  no  corresponding  point  at 
the  other  end  of  the  line,  and,  in  order  to  obtain 
a  check,  we  draw  the  line  PCsP?  and  compute  its 
bearing  and  length,  to  be  run  in  the  field  as  an 
auxiliary  line  to  check  on  point  PT. 

CHECK  POINT  A.— We  first  write  the  equa- 
tions for  lines,  P2Ps  and  PiVi.  The  general  equa- 
tion for  a  straight  line  is  y  =  ax  +  b, 

where  a  =  (yn  —  y^  /  (xn  —  xk) 

and  b  =  yk  —  axk. 

(The  subscripts  k  and  n  refer  to  the  initial  and 
final  points  respectively  of  any  line.) 

9 


For  line  Pg  P3  a  =  (y3  —  y2)  /  (x3  —  x2)  —  (1,700  —  600) 
(2,600  —  1,600)  =  1,100/1,000  =  1.1;  and  b  =  y,  —  ax2  = 
600  —  1.1  (1,600)  =  —  1,160.  Equation  of  line  PaP8is 
therefore, 

y=l.l  x  +  (—1,160) 
or  y  -  1.1  x  —  1,160. 


For  line  PiVi 


i,  6W—0  _  04.954 

3,250-0 


and 


making  the  equation  of  PiVj 

y  =  0.4954  x. 

We  now  compute  the  co-ordinates  of  the  point  A 
of  intersection  of  PaP2  and  PiVi. 

If  the  equation  of  the  first  line  be  written 

y  =  ax  +  b 
and  the  equation  of  the  second  be  written 

y  =  a'  x  +  b' 
then  the  co-ordinates  of  this  common  point  are  : 


and  a  check  is  had  in  the  equation 

y  ~a!x>  +V 

Substituting  numerical   values  of   a,   b,  a',  b',  for   rou 
point  A,  we  get  : 


(-  J.160J  -  951 
We  scale  the  map  to  get  a  rough  check  on  the  computations. 

It  remains  to  find  for  A  the  station  and  plus 
via  each  line. 
The  distance 


log         319  =  2.50379 
log  sin  42°  16'  =  9.82775 

log     474.8  =  2.67604 
The  logarithmic  computation  makes  P2  A  —  474.3. 

10 


The  station  and  plus  of  A  on  the  preliminary  Is 

PI  P2  -f  ?2  A.  =  1,709  +  474.3  -  21  +  83.3. 
The  distance 


as  computed  by  logarithms  here  : 

log    1,919  =  3.28299 
log  sin  63°  39'  =»  9.95236 

log    2,141  =  3.330Q3 

The  station  and  plus  for  A  on  the  location  is, 
then,  21  +  41. 

Scale  the  map  to  roughly  check  these  values. 

It  is  evident  that  station  21  +  41  on  the  location 
survey  should  coincide  with  station  21  +  83.3  on 
the  preliminary. 

In  cases  in  practice  a  check  point  may  be  con- 
veniently obtained  by  producing  a  location  tan- 
gent to  intersect  the  preliminary,  and  making  the 
computation  in  the  foregoing  manner. 

CHECK  LINE  PCaF?.—  When  a  crossing  of  the 
location  with  the  preliminary  is  not  near  at  hand 
for  a.  desired  check  on  the  field  work,  a  check 
line,  or  tie  line,  is  drawn  between  a  chosen  point 
of  the  location  and  a  chosen  point  of  the  pre- 
liminary; and  the  bearing  and  length  of  the  check 
line  are  computed.  As  an  example,  draw  the 
check  line  PCsP?,  and  find  the  bearing  by  the 
equation: 


Vr,  -ypc3        4,500-4,120 
and  find  the  length 


log    470  —  2.67210  log    470  —  2.67210 

log    380  =  2.57978        log  sin  51°  03'  =  9.89081 

log  tan  51o  03'  =  0.09232  log    604.4  =  2.78129 

We  find  the  bearing  and  length  of  PCaP?  to  be 
N  51°  03'  E,  604.4  ft.  The  location  having  been 
carried  in  the  field  to  the  point  PCs,  the  transit- 
man  deflects  to  the  left  at  this  point  the  angle 

^PCs   -  ^PCsPr   =   78°   36'  ~  51°   °3'   ----    27°   33/' 
and  the  chainmen  lay  off  604.4  on  this  course,  and 

11 


should  by  so  doing-  arrive  precisely  at  PT  on  the 
preliminary. 

10.  COMPUTE  CURVE  DEFLECTIONS.—  PCi  is 
22  +  72  making-  the  first  sub-chord  on  the  6-degree 
curve,  28  ft.  The  corresponding  deflection  is  .28  x 
3°  =  .28  x  180'  ==  50',  i.  e.,  the  tangent  deflection 
for  station  23  is  50  minutes.  The  deflection  for 
sub-chord  at  .PCa  is  .99  x  240'  =  238'  =  3°  58'. 
The  deflection  for  sub-chord  at  PTi  is  .99  x  180'  = 
178'  =  2°  58'.  The  deflection  for  sub-chord  at 
PT8  is  .58  x  24tf  =  139'  =  2°  19'. 


ALGORITHMS.—  To    facilitate    explanation    the 
foregoing  computations  have  been  put  down  in  a 


Red- 


Fig.    2.     Design    for    Rulings    on    Computing    Paper. 

rambling  manner  which  makes  the  computed 
quantities  hard  to  find  when  wanted.  For  a  lon- 
ger location  than  this,  economy  of  time  and  effort 
requires  that  the  routine  computations  be  syste- 
matically arranged,  that  like  operations  as  well  as 
like  quantities  may  be  brought  together.  By  this 
means  entering  the  data  and  making  the  computa- 
tions in  a  short  time  become  largely  mechanical 
processes,  and  at  the  end  the  computed  quantities 
stand  in  tabular  order  and  may  be  quickly  found 
when  wanted.  For  the  computations  of  this  ar- 


tide  the  following  computation  forms,  or  algo- 
rithms, are  suggested.  While  these  may  not  pre- 
sent the  best  arrangement,  they  will  at  least  show 
the  advantage  of  order  over  disorder.  The  reader 
may  be  interested  in  the  fact  that  the  writer's  stu- 
dents buy  for  their  computing,  letter-size  sheets 
of  paper  ruled  on  one  side  with  the  special  design 
shown  in  Fig.  2.  By  the  use  of  this  paper  any 
algorithm  may  be  followed  without  drawing  lines, 
and  columns  of  digits  are  kept  vertical  automat- 
ically. 

Algorithm  1  is  for  computing  the  bearing  and 
length  of  lines  which  are  terminated  by  points  of 
known  co-ordinates.  Column  A  contains  the  num- 
bers of  the  horizontal  lines  of  the  algorithm.  Col- 
umn B  contains  the  symbols.  Xk,  yk  designate  the 
co-ordinates  of  the  initial  point  of  any  line;  xn,  yn 
designate  the  co-ordinates  of  the  final  point  of  the 
line;  )3kn  is  the  bearing,  and  dkn  is  the  length  of 
the  line.  The  lower  half  of  this  algorithm  has  to 
do  with  the  check  equation: 

hypot.  =  (base2  +  altitude2)1/2. 

Notice  that  the  values  on  line  21  check  those  on 
line  12.  Of  course  this  check  will  not  detect  errors 
made  in  entering  values  of  co-ordinates  on  lines 
1,  2,  3,  4. 

When  a  table  of  squares  is  at  hand  it  may  well 
be  used  in  place  of  logarithms  for  the  computation 
of  checks.  Columns  C,  D,  E,  F  give  for  the  line 
Pi  Vi,  Vi  Va,  Va  PCs,  and  PCsP?  respectively,  the 
numerical  quantities  corresponding  to  the  symbols 
of  Column  B.  It  is  suggested  that  the  order  of 
steps  in  computing  be:  (1)  Enter  the  co-ordinates 
for  all  the  lines  concerned;  (2)  Set  down  on  lines 
5  and  6  the  differences;  (3)  Enter  all  the  logs  on 
line  7;  (4)  Enter  all  the  logs  on  line  8,  and  so  on; 
every  operation  in  Column  C  being  immediately 
repeated  for  the  succeeding  columns. 

Algorithm  2:  Column  A  contains  the  numbers 
which  have  been  used  to  designate  the  curves.  In 
B  will  be  found  the  degree  of  each  curve.  Each 
degree  of  curve  is  repeated  in  parenthesis,  with 
the  minutes  expressed  in  decimal  of  a  degree.  In 
C  are  entered  the  central  angles,  each  of  which 
has  been  computed  from  two  values  of  £  taken 
from  Algorithm  1.  For  example:  C2  (i.  e.,  the 

13 


ICC*'* 

CO  COO 

_,   ^ibw^wS®      ^ 

^    CO  T-J  iM  iH  CO  kO  l> 

ko  kOko  koc* 


2s: 


OoOo 


«rHOkOr-iCO.SlOCS  CO  CO<N-*  CD  rHTjt 
>>  •H-^OS-^iMCD'H  k(5  iHkOiM  CO  COC^ 
<D  l^»  05  0500 


CO'M'O'OS'CO'      t-coio 


l  O  O  C<l 


t- 


^  OOOOOO  L-  ^^^ 

O7  kO  rH  kO  r-l  00  CO    kC  CD  Cq  (M  -  CD  CO  O  O  CS  ' 

iMCD<NCDOOOO   O  COkCCO  ^5  t-«DCDC<lkOo»*  — 

®  CO  r-l  CO  rH  rH  CD    ifl  C^  05  CO  CO  CO  CO  kO  05  iH  05  05  CO 

«  rHO    O  kOkO  Q  C*  rH  O  kO  CO  r-l  kO 

eoco  o'  oico  ®  »*co  t-co 


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00    O    00 


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:  :  :  :3 


rH  <N  CO  "*  kO  CD  t-  00    05    OrH(M 


14 


-9p  paoqoqng 


PITB  "BIS 


snid  pire  " 


iu    •        -i     .     .     .      , 

o    .to   :  :  :  M 

01     •     <S1     •     .     . 


Z  jo  urns 

•jstp  -Sum  I      S 

H      -UOT!J09e.T9!lUI 

•g     05  p9onpojd    ^  c^ 

4  (ITTQ^TTTJI  CD 


iS  i  i|  : 


3" 


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:g::s:::« 
ifi  :  '•%  :  ':  :  « 


<]   r 


15 


quantity  in  Column  C  and  on  line  (2)  is  109°  37', 
which  is  the  sum  of  C13  (Alg.  1)  and  D13  (Alg.  1). 
C3  and  C6  give  the  central  angle  with  minutes  re- 
duced to  decimal  of  a  degree.  D  contains  the 
tangent  distance  for  each  curve.  E  contains  the 
curve  lengths.  In  P  are  entered  the  lengths  of 
tangents  produced  to  intersection  points,  copied 
from  line  12  of  Alg.  1.  G  contains  the  amounts  to 
be  subtracted  from  quantities  in  F  to  obtain  the 
quantities  in  H.  For  example:  G4  is  the  sum  of 
D2  and  D5,  and  H4  is  F4  less  G4.  I  contains  the 
station  and  plus  for  each  point  of  curve;  and  J 
contains  the  station  and  plus  for  each  point  of 


ALGORITHM  III.— Line  Equations. 


Formulas  :  a  =  (yn  — 


=  yk  — 


xn 


Line  P2  P3  Line  PiVj 

1600  0 

600  5 

2600  3250 

4 yn                  1700  1610 

5 yn  —  yk                 1100  1610 

6 xn  —  xk                  1000  3250 

7 log(yn-yk)                    3.04139  3.20683 

8 log(xn  — xk)                     3.00000  3.51188 

9 log  a                    0.04139  9.69495 

10 a                         1.1  0.4954 

11 logxk                    3.20412  

12 logaxk                    3.24551  

13 axk                 1760  0 

14 b             —1160  0 

15 (Eq.)y  =  ax  +  b      y  =  l.lx  — 1160  y  =  0.4954x  +  0 

ABC 

ALGORITHM  IV.— Co-ordinates  of  Intersection  Point. 

Formulas  :    x   =    (b'   —   b)    /   (a   —   a')  ;    y    =   ax    +    b ; 
check,  y  =  a'x  +  b'. 

Check  Point  A. 

1..., (Eq.iy  =  ax  +  b  y=l.lx— 116O 

2 (Eq.)'y  =  a'x  +  b'  y=  0.4954  +  0 

3 a  1.1 

4 a'  0.4954 

5 b  —1160 

6 b'  0 

7 b'— b  1160 

8 B—  a'  0.6046 

9 log(b  — b)  3.06446 

10 log  (a  — a'  9.78147 

11 log              x  3.28299 

12 log              a  0.04139 

13 log             a'  9.69495 

14 log            ax  3.32438 

15 log          a'x  2.97794 

16 ax  2110. 

17...                                                            a'x  950.5 

18 x  1919. 

19 y(=ax  +  b)  950. 

20 y(=a'x  +  b')  950.5 


1G 


tangent.     12  is  rewritten  from  HI;  J2  is  12  plus 
E2;  15  is  J2  plus  H4;  J5  is  15  plus  E5,  and  so  on. 
K  and  L  contain  the  subchord  deflections. 
Algorithms  3  and  4  will  need  no  explanation. 

11. — ALINEMENT  NOTES  FOR  LOCATION. — 
We  are  now  ready  to  enter  the  results  of  our  com- 
putations in  the  field  book  in  the  usual  form  of 
alinement  notes. 

Location  Alinement  Notes. 
Computed 

Sta.    Curve  Point.    Deflections.     Bearings. 
69  +  22       PC8  Check  :    Deflect  27°  33'  (bearing  N  51° 

03'  E)  to  left  and  run  604/4  to  P7  which 
....  is  Sta.  84  +  26  on  preliminary. 

60 

59  N  78°  36'  E 

58  +  58        PT2  62°  17'  N  16°  20'  E 

58  59°  58' 

57  55°  58' 


45  7°  58' 

44  3°  58' 

43  +  01  PC2  8°  R     0°  00'           N  45°  58'  W 

43 

42 

41  N  46°  58'  W 

40  +  99  PT!               54°  48.5'       N    8°  51'  E 

40  51°  50' 

39  48°  50' 


25  6°  50' 

24  3°  50' 

23  50' 

22  +  72  PCi   6°  L        0°  00'          N  63°  39  E 

22 

21  +  41  Check  :  This  point  is  21  +  83.3  on  preliminary. 

21 

"2" 

1 

0  This  is  Sta.  0  of  preliminary.    For  first  location 

course  deflect  5°  48'  to  left  of  first  preliminary 

course . 

THE  MATHEMATICS  OF  PAGPER  LOCATION : 
USING  POLAR  CO-ORDINATES.-^To  compute 
the  location  field  notes  without  employing  rect- 
angular co-ordinates,  proceed  thus: 

1.  The  location   having  been   laid   down   on  the 
map  as  shown,  scale  PiVi,  ViV2,  "WFCa,  and  with 
the  protractor  scale   the  bearings  of   these   three 
lines. 

2.  OomDUte  the  central  angles  A  i  and  £«. 
8.  Compute  curve  lengths  La.  and  L2. 

4.  Compute  tangent  distances  Ti  and  Tz. 
5.  Compute  lengths  of  tangents  PiPCi,  PTi    PCs 
and  PTaPCs. 

17 


6.  Write  station  and  plus  for  Pi,  PCi,  PTi,  PC», 
PTa,  and  PC8. 

7.  Find  preliminary  station  and  plus,  and  loca- 
tion station  and  plus,  for  check  point  A,  thus:  (a) 
Compute  angles  of  triangle  PilPsA;   (b)   Prom  the 
known  side  and  angles  of  this  triangle  compute 
the  sides  PiA  and  P2A;    (c)  The  preliminary  sta- 
tion and  plus  for  A  is  PiP2  +  P^A;  and  the  loca* 
tion  station  and  plus  for  A  is  PiA. 

8.  Having   drawn    on    the    map    the   check    line 
PCsP?,   compute  the  length   and  bearing  'of    this 
line.    This  is  done  by  treating  PCaP?  as  the  "miss- 
ing  side"    of    the    closed   figure    APaP^PriPftPTPCs- 

VaViA. 

9.'  Write  the  alinement  notes  in  the  field  book. 

The  work  of  finding  the  "missing  side"  in  Step  8, 
just  above,  involves  practically  all  the  operations 
required  to  compute  the  co-ordinates  of  the  chief 
points  of  our  map  from  the  field  notes  of  the  pre- 
liminary survey. 

Computing  from  the  field  notes  the  rectangular 
co-ordinates  of  the  chief  points  of  a  survey  is 
analogous  to  computing  from  the  level  notes  the 
elevations  of  the  stations  on  a  profile,  and  has 
similar  advantages. 


18 


O  S^\  f\  IS  C     on    any   subject   in    which 
you  may  be  interested  sup- 
plied promptly  on   receipt  of  price.     Send 
for  catalogue. 

M.    C.    CLARK, 

PUBLISHER  AND  BOOKSELLER, 
13-21  PARK  Row, 

NEW  YORK. 


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) 

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